Problem: Simplify; express your answer in exponential form. Assume $x\neq 0, p\neq 0$. $\dfrac{{(x^{-2}p^{-2})^{-2}}}{{(x^{-3}p^{4})^{2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{-2}p^{-2})^{-2} = (x^{-2})^{-2}(p^{-2})^{-2}}$ On the left, we have ${x^{-2}}$ to the exponent ${-2}$ . Now ${-2 \times -2 = 4}$ , so ${(x^{-2})^{-2} = x^{4}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{-2}p^{-2})^{-2}}}{{(x^{-3}p^{4})^{2}}} = \dfrac{{x^{4}p^{4}}}{{x^{-6}p^{8}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{4}p^{4}}}{{x^{-6}p^{8}}} = \dfrac{{x^{4}}}{{x^{-6}}} \cdot \dfrac{{p^{4}}}{{p^{8}}} = x^{{4} - {(-6)}} \cdot p^{{4} - {8}} = x^{10}p^{-4}$